Question

A film class had 33 students who liked Hitchcock movies, 21 students who liked Spielberg movies, and 17 students who liked both kinds of films. How many students were in the class if every student is renresented in the survey?

Short answer

There are 37 students in the class.

Step-by-step solution

Identify Given Values

Begin by identifying the provided values from the problem. We know there are 33 students who like Hitchcock movies, which we denote as \( |A| = 33 \), 21 students who like Spielberg movies, denoted as \( |B| = 21 \), and 17 students who like both Hitchcock and Spielberg movies, denoted as \(|A \cap B| = 17\).

Use Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle provides a way to find the number of students who like at least one type of movie. The formula is: \( |A \cup B| = |A| + |B| - |A \cap B| \). Substitute the values: \( |A \cup B| = 33 + 21 - 17 \).

Calculate Total Students

Compute \( |A \cup B| \) using the formula from the previous step. Perform the arithmetic: \( |A \cup B| = 33 + 21 - 17 = 37 \). This means that there are 37 students who like at least one type of movie, which is the total number of students surveyed, given that every student is represented.

Key concepts

Sets and Subsets

Imagine sets as collections of items or elements. In mathematics, sets can represent any group of things or numbers.
In our film class example, we have a set of students who like Hitchcock movies and another set for those who like Spielberg movies. We also have a subset of students who enjoy both.

• Set A: Students who like Hitchcock movies.
• Set B: Students who like Spielberg movies.
• Subset of both: Students liking both Hitchcock and Spielberg movies terms overlaps, known as intersections, in set theory.

The task in the exercise involves exploring these unions and intersections using the principle of inclusion-exclusion, an essential concept in set theory. Understanding how sets combine or intersect provides a clearer picture of complex relationships among grouped elements.

Problem Solving

Problem solving in mathematics often involves applying known strategies to compute unknown quantities. An effective way is to first identify the problem and the information given.
In our example, the known quantities are:

• 33 students like Hitchcock movies.
• 21 students like Spielberg movies.
• 17 students like both.

To solve the problem of finding the total number of students, you need to recognize that some students are counted twice — once in each category of fans. This is where the Inclusion-Exclusion Principle becomes pivotal. It helps determine the exact amount by preventing double-counting those who belong to multiple categories.
Using the formula \(\left| A \cup B \right| = \left| A \right| + \left| B \right| - \left| A \cap B \right|\), we calculate the total number of unique elements in the union of two sets, ensuring effective problem resolution.

Hitchcock and Spielberg Movies

Hitchcock and Spielberg are both iconic directors, each with a distinct style that attracts fans across generations. This exercise contextualizes mathematics within the realm of film fandoms.
Considering their significance in the film industry, it is not surprising that many students enjoy their movies. Alfred Hitchcock, known for his suspense thrillers, appeals to those who enjoy mystery and tension. In contrast, Steven Spielberg, recognized for his adventure and science fiction films, draws fans who prefer thrilling narratives and visual effects.
Understanding the overlap between the students in this exercise not only provides insights into their movie preferences but also events highlights societal intersections of different cultural tastes, which can be mathematically represented. Breaking down the overlap explains the common ground and combined appeal both directors have among audiences.